Modelling chemistry and biology after implantation of a drug-eluting stent. Part Ⅰ: Drug transport

Figure 1.  The deployment of a drug-eluting stent in a diseased coronary artery. The stent is coated with a drug-loaded polymer, and subsequent to deployment of the stent, drug releases from the stent coating into the artery wall to prevent re-blockage due to restenosis

Figure 2.  Drug diffuses from a polymer coating into the artery wall. In the arterial tissue, drug molecules can associate with and dissociate from specific binding sites, and can diffuse in their free form. Drug molecules may also be convected by the outward movement of plasma through the artery wall.

Figure 3.  Numerical solutions to the initial boundary value problem (5-7) for various dimensional times $t=O(L_a^2/D_a)$, the diffusion time scale for free drug in the artery wall. The drug penetrates the artery wall on this time scale and this is evident in the figures. We have plotted profiles for the bound drug in the artery wall in (a), and the free drug in the polymer and the artery wall in (b). The parameter values used are $L=60, \varepsilon=10^{-6}$, $\eta = 0.002, K_b = 1700$, and $Pe=0.2$

Figure 4.  Numerical solutions to the initial boundary value problem (5-7) for various dimensional times $t=O(L_p^2/D_p)$. On this time scale, the behaviour in the arterial tissue is quasistatic, and is driven temporally by the decreasing flux of drug from the stent coating. We have plotted profiles for the bound drug in the artery wall in (a), and the free drug in the polymer and the artery wall in (b). The parameter values used are $L=60, \varepsilon=10^{-6}$, $\eta = 0.002, K_b = 1700$, and $Pe=0.2$

Figure 5.  Plots of the average bound drug, $m_a(t)$, in the artery wall as a function of time $t$ for various values of $\eta$, and with $L=60$, $\varepsilon=10^{-6}, K_b = 1700$, and $Pe=0.2$. On the vertical scale, $1$ corresponds to full occupancy of the specific binding sites. The rapid rise in the profiles near $t=0$ corresponds to the short time scale over which free drug in the tissue crosses the artery wall. In (a), $m_a(t)$ was calculated from numerical solutions of the initial boundary value problem (5-7). It is seen that there is significant occupancy of the binding sites for a period of a few months subsequent to the stent being implanted. In (b), we compare the asymptotic solution (28) to numerical results for the first month. The sub window shows the relative errors between the asymptotic and numerical solutions, $(m_{a,\text{numerical}}-m_{a,\text{asymptotic}})/m_{a,\text{numerical}}$

Figure 6.  Comparison between in-vivo experimental data and modelling profiles for the fraction of the total drug released. In (a), Rapamycin release from Cypher [24] with $D_p=1.2 \times 10^{-10}$ mm$^2$/s and the mean squared error MSE $= 9 \times 10^{-4}$. In (b), Everolimus release from Xience V [20] with $D_p= 1.1 \times 10^{-11}$ mm$^2$/s and MSE $= 9 \times 10^{-3}$

Figure 7.  Comparison of the average mass of drug in the artery wall calculated by (30) to in-vivo experimental data after (a) implantation of Cypher stents from 1 to 30 days [24] and (b) implantation of Xience V stents from 1 to 120 days [20]. Values have been normalized with respect to the initial drug content, $M(0)$. The parameter values used are: $L_a=0.45$ mm, $D_a=2.5 \times 10^{-4}$ mm$^2$/s, $V_a=5.8 \times 10^{-5}$ mm/s, $K_b = 300$; (a) $L_p=1.26 \times 10^{-2}$ mm, $M(0)= 174.89$ $\mu$g, $D_p=6.0 \times 10^{-11}$ mm$^2$/s, $\eta = 0.0075$ (MSE=$1.7 \times 10^{-4}$); and (b) $L_p= 7.6 \times 10^{-3}$ mm, $M(0)= 100$ $\mu$g, $D_p=1.0 \times 10^{-11}$ mm$^2$/s, $\eta = 0.002$ (MSE=$2.2 \times 10^{-4}$)